Oscillator strengths of defects in insulators : The generalization of Smakula's equation

A generalization of Smakula's equation is developed for the case of defect absorption in host materials with weak absorption but rapidly varying refractive index. As an example the generalization is applied to the a and /3 bands in ionic crystals. In the case of additively-colored KI it is found that the ratio of oscillator strengths is x 0.6. This result is in qualitative disagreement with the commonly accepted theoretical treatment which for NaCl predictsf,/f, x 2. Possible reasons for this disagreement are discussed. 1 . Introduction. Smakula's [l] celebrated relation for the strength of a defect's optical absorption has formed the basis of quantitative absorption studies for fifty years. In the commonly used Mollwo-Roos [2] form, Smakula's equation states that for defects with oscillator strength f and density p in a host medium of refractive index no where ,urnax and r are the maximum absorption coefficient and the full width at half maximum of the defect's absorption band, respectively. This relation was originally deduced for a dilute solution of absorbing species in a transparent medium with constant refractive index on the assumption that the absorption is Lorentzian and that the field at the absorbing centre is the Lorentz local field. A less restrictive derivative leads to the more modern form [3, 41 where (Geff/G0) is the ratio of the effective field at the In many current studies, two of the key assumptions made above do not hold : 1) The host medium is often not completely transparent, and 2) The refractive index of the host is not constant over the defect's absorption band. Examples of both of these conditions occur for defect or perturbed-exciton absorptions just to the long-wavelength side of the fundamental absorption of the host medium. In the present paper we generalize Smakula's equation by relaxing these assumptions and we apply the results to the aand P-band absorptions in the alkali halides as an example. 2. Generalization of Smakula's equation. In essence Smakula's equation is a partial f sum rule. However, care is necessary in its generalization because the f sum rule can be written in several equivalent forms when the sum over the entire spectrum is taken [5]. The forms of interest here are for the imaginary part of the dielectric function, E,, absorbing defect the average in the medium and for the imaginary part of the refractive index, n(w), and no assumption is made about the absorption band shape. (*) Work supported by the U.S. Department of Energy and the Deutsche Forschungsgemeinschaft. where N is the electron density. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980621 OSCILLATOR STRENGTHS O F DEFECTS IN INSULATORS C6-8 1 While &,(o) and x (o ) obey these very similar j sum rules over the whole spectrum, their partial f sums over a finite-energy interval differ markedly. (For an example, see Ref. [5].) This difference is obvious if one recalls that E ~ ( O ) = 2 n(o) ~ ( c o ) . Physically it can be seen from the fact that E ~ ( C U ) is a measure of heat generation, while x(w) is related to the decrease in amplitude of the transmitted light wave. In the present context the quantity of interest is the dipole oscillator strength which is directly related to the polarizability and hence to c2. Thus, eq. (3) is the appropriate form of the f sum to consider over the finite-energy interval a d o < b. This leads to where f is the oscillator strength ; that is, the fraction of electrons effective for absorption in the frequency range a < o d b. In optical experiments %(a) , not E ~ ( c o ) , is generally measured so that there is a temptation to define the oscillator strength from eq. (4) rather than eq. (3) to avoid having to have a knowledge of n(co). However, this is incorrect because it leads to the omission of the factor of n(o) found in the second form of eq. (5). The importance of this factor is illustrated in figure 1 which shows the absorption of a model system consisting of a single strong Lorentz oscillator representing the host material and a weaker oscillator representing the imperfection the latter was chosen to have an oscillator strength times that of the host corresponding to the case of a dilute imperfection. The parameters were chosen so that the host refractive index, no(o), is approximately 1.5 over most of the region of transparency, but rises rapidly just below the host absorption frequency, o,,,. The imperfection absorption frequency was varied over the ranges 0.3 o , , d o < 0.9 o,,. Despite the fact that the oscillator strength of the imperfection remained a constant, the imperfection absorption band appears to become weaker and very distorted as it approaches the tail of the host absorption. Qualitatively this can be viewed as a result of the shielding of the imperfection by the polarizable host (I). The factor n(w) in eq. (5b) compensates the absorption spectrum as measured by x ( o ) for this shielding. The importance of refractive index variation for the intrinsic absorption of a semiconductor was noted previously by Dexter [5a]. ( ') Detailed considerat~ons show that a compensatory increase in the host absorption occurs in another spectral region so that the f sum rule for 40) over the entire spectrum (eq. (4)) remains valid when both host and imperfection spectra are included in the summation. DEFECT ABSORPTION BANDS FOR VARIOUS TRANSITION ENERGIES Fig. I. -The absorption spectra of a model system consisting of a weak defect absorption on the low-energy side of a stronger absorption that represents the fundamental crystal absorption. Spectra for four possible defect absorption energies, cud = ~ o , , ~ , are shown. Although the oscillator strength of the defect is held constant, the defect absorption appears to become weaker for higher values of v because of the shielding effect of the polarizable host medium. In the defect problem the quantity of primary interest is the change in oscillator strength introduced by an imperfection. The resulting partial f sum for the defect can be derived in an approximate form by taking differentials of eq. (5) or without approximation by applying superconvergence techniques [6] to the function ~ k / ~ ( o ) ~ ~ / ~ ( o ) ~ ~ ( o ) , where &,(o) is the dielectric function of the unperturbed host and &(a) is the'dielectric function of the host material with defects. The result is : where An and Ax are the changes in the real and imaginary parts of the dielectric function caused by the defects. Here no(o) and xo(w) are the real and imaginary parts of the refractive index of the unperturbed host crystal. In eq. (6) the ratio of the average field in the medium, Go, to the effective field at the centre, Geff, has been included to take into account the possibility that the local field acting on the defect may differ from the C6-82 D. Y. SMITH AND G. GRAHAM average field in the medium [3]. In this case the oscillator strength is that which the defect would exhibit in free space. For a host substance which is only weakly absorbing in the range a < o < b this expression reduces to where Ax is just the defect-induced change in absorption. In the event no(o) is a constant, this reduces to eq. (2) since p(o) = 2 ox(o)/c. In transmission experiments, eq. (7) is generally a sufficient approximation to eq. (6) since xo(o) must be small in order that transmission is possible in the first place. The opposite, of course, holds in a highly absorbing region where the second term of eq. (6) dominates the oscillator strength. 3. An example : the aand /3-bands. As an example of the case of the present formalism, the generalized Smakula's equation was applied to the spectra of the aand P-bands [7] in KI. The observed spectra of an additively colored sample measured at liquid nitrogen temperature are shown in figure 2. OPTICAL ABSORPTION OF ADDlTlVELY COLORED K I AT 77K I I